Divisibility rules for Numbers with examples

Divisibility Rules for Numbers

Divisibility by 2

-> A number is divisible by 2 if and only if the last digit is divisible by 2.
Example: Not required :)



Divisibility by 3
-> A number is divisible by 3 if and only if the sum of the digits is divisible by 3.
Example: 
Which of the following two numbers is/are fully divisible by 3: 97533222 or 97533322?
Answer:  97533222
Reason:
Sum of digits for 97533222 =   33 / 3 = 11
Sum of digits for  97533322 =   34 / 3 = 11.33  

Divisibility by 4
-> A number is divisible by 4 if and only if the last 2 digits is a number divisible by 4.
Example: 
Which of the following two number is fully divisible by 4: 5555444446 or  5555444448 ?
Answer:  5555444448
Reason:
Last 2 digits for  5555444446  are 46, 46 is not fully divisible by 4 (46/4 = 11.5)

Last 2 digits for  5555444448  are 48, 48 is fully divisible by 4  (48/4 = 12)

Divisibility by 5
-> A number is divisible by 5 if and only if the last digit is divisible by 5.
Example: If a number ends in 5 or 0 then it would be divisible by 5. Such as 777995, 13170



Divisibility by 6
-> A number is divisible by 6 if and only if it is divisible by 2 and 3.
Example: 
Which of the following numbers is/are fully divisible by 3: 97533222, 97533322, 97533225, 97533228?
Answer:  97533222 and 97533228
Reason:

For a number to be fully divisible by 6 it should satisfy following two conditions:

1. It must be a even number (which means out of 4 options we can decide that 97533225 is surely not divisible by 6 as it is not an even number)

2. It must be divisible by 3(sum of all digits of particular even number should be divisible by 3)

Sum of digits for 97533222 = 33 (fully divisible by 3)

Sum of digits for 97533322 = 34 (not fully divisible by 3)

Sum of digits for 97533228 = 39 (fully divisible by 3)

Divisibility by 7

-> To find out if a number is divisible by seven, take the last digit, double it, and subtract it from the rest of the number, if the resulting number is divisible by 7 then original number is divisible by 7. If you don't know the new number's divisibility, you can apply the rule again.

Example: 
Which of the following numbers is/are fully divisible by 7:  203, 3192, 3197, 38241?

Answer : 203, 3192, 38241

Reason:

For 203, you would double the last digit to get six, and subtract that from 20 to get 14. Hence 203 is divisible by 7.

Applying the same rule to check if 3192 is divisible by 7:

3192 => 319 - 2*2 = 315 => 31 - 2*5 = 21, 21 is divisible by 7  hence 3192 is divisible by 7

Applying the same rule to check if 3197 is divisible by 7:

3197 => 319-2*7 = 305 => 30 -2*5 = 20, 20 is not divisible by 7 hence 3197 is not divisible by 7

Applying the same rule to check if 38241 is divisible by 7:

38241 => 3824 -2*1 = 3822 => 382 - 2*2 = 378 => 37 -2*8 = 21, 21 is divisible by 7  hence  38241  is divisible by 7

Divisibility by 8
-> A number is divisible by 8 if and only if the last 3 digits of a number divisible by 8.
Example: 
Which of the following numbers is/are fully divisible by 8: 97533224, 97533328, 97533222, 97533228?
Answer:  97533224 and 97533328
Reason: 224 and 328 are divisible by 8 whereas 222 and 228 are not divisible  by 8.



Divisibility by 9
-> A number is divisible by 9 if and only if the sum of the digits is divisible by 9.
Example: 
Which of the following numbers is/are fully divisible by 9: 111111111, 111222, 2222244, 66669 699999?
Answer:   111111111, 111222  and 2222244
Reason: Sum of digits for 111111111, 111222  and 2222244 are 9, 9 and 18 respectively all of which are divisible by 9. Whereas sum of digits for 66669 & 699999 are 33 & 51 respectively both of which are not divisible by 9. 



Divisibility by 10
-> A number is divisible by 10 if and only if the number ends in n zeros.
Example: If a number ends in 0 then it would be divisible by 5. Such as 111110, 876543200  


Divisibility by 11
-> A number is divisible by 11 if the difference of the sum of digits at odd places and the sum of its digits at even places, is either 0 or divisible by 11, then clearly the number is divisible by 11.

Example: 
Which of the following numbers is/are fully divisible by 11:  6050, 3035362, 90002, 900002?
Answer:   6050, 3035362, 90002
Reason:   Sum of digits at even place in 6050 = 11 and sum of digits at odd place is 0. Difference between odd and even = 11. Hence 6050 is divisible by 11.
Similarly sum of digits at even place in 3035362 = 11 and sum of digits at odd place is 11. Difference between odd and even = 0. Hence 3035362 is divisible by 11.
Similarly Difference between odd and even digits for 90002 is 11 whereas for 900002 difference between odd and even digits is 7. Hence 90002 is divisible by 11 but 900002 is not.




Divisibility by 12
-> A number is divisible by 12 if the number is divisible by both 3 and 4
Example: 
Which of the following two numbers is/are divisible by 12: 22584 and 22756?
Answer: 22584
Reason: 22584 is divisible by 3 because sum of all digits =21 which is divisible by 3 and 22584 is also divisible by 4 because last 2 digits of 22584 i.e. 84 is divisible by 4. Hence 22584 is divisible by 12
22756 is not divisible 3 as sum of digits are 22 which is not divisible by 3.




Divisibility by 13
-> To find out if a number is divisible by 13 take the last digit of number and multiple it by 4 and add it to number formed with remaining digits check if the resultant number is divisible by 13. In case resultant number is big repeat process of multiplying last digit by 4 and adding it to number formed by remaining digits.
Example: 
Which of the following numbers is/are divisible by 13: 11013, 110006, 110026 ?
Answer: 110006
Reason: Applying the rule above of multiplying last digits by 4 and adding it to number formed by remaining digit we get : 1100+4*4= 11024
Continuing the same way with resultant number we get : 1102+4*4= 1118
=> 1118 = 111+8*4 = 143
=> 14+3*4 = 26. 26 is divisible by 13. Hence 110006 is divisible by 13
11013 is not divisible by 13 as per results below:
11013= 1101+3*4 = 1113 => 1113 = 111+3*4 = 123 => 123 = 12+3*4 =24
24 is not divisible 13 hence 11013 is not divisible by 13.

-> If n is even , n(n+1)(n+2) is divisible by 24


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